\(C^*\)algebras of Toeplitz type associated with algebraic number fields
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DOI: 10.1007/s0020801208269
 Cite this article as:
 Cuntz, J., Deninger, C. & Laca, M. Math. Ann. (2013) 355: 1383. doi:10.1007/s0020801208269
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Abstract
We associate with the ring \(R\) of algebraic integers in a number field a C*algebra \({\mathfrak T }[R]\). It is an extension of the ring C*algebra \({\mathfrak A }[R]\) studied previously by the first named author in collaboration with X. Li. In contrast to \({\mathfrak A }[R]\), it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the \(ax+b\)semigroup \(R\rtimes R^\times \) on \(\ell ^2 (R\rtimes R^\times )\). The algebra \({\mathfrak T }[R]\) carries a natural oneparameter automorphism group \((\sigma _t)_{t\in {\mathbb R }}\). We determine its KMSstructure. The technical difficulties that we encounter are due to the presence of the class group in the case where \(R\) is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMSstates splits over the class group. The “partition functions” are partial Dedekind \(\zeta \)functions. We prove a result characterizing the asymptotic behavior of quotients of such partial \(\zeta \)functions, which we then use to show uniqueness of the \(\beta \)KMS state for each inverse temperature \(\beta \in (1,2]\).
Mathematics Subject Classification (2000)
Primary 22D2546L8911R0411M551 Introduction
 (a)
The \(u^x\) and the \(s_a\) define representations of the additive group \(R\) and of the multiplicative semigroup \(R^\times \), respectively, (i.e. \(u^xu^y=u^{x+y}\) and \(s_as_b=s_{ab}\)) and moreover we require the relation \(s_a u^x=u^{ax}s_a\) for all \(x\in R,\,a\in R^\times \) (i.e the \(u^x\) and \(s_a\) together give a representation of the \(ax+b\)semigroup \(R\rtimes R^\times \)).
 (b)
For each \(a\in R^\times \) one has \(\sum _{x\in R/aR}u^xs_as^*_au^{x}=1\).
Now, to obtain a presentation of \(\mathfrak T [R]\) we essentially have to relax, in this presentation of \(\mathfrak A [R]\), condition (b) to the weaker condition \(\sum _{x\in R/aR}u^xs_as^*_au^{x}\le 1\). This modification in the relations is sufficient to characterize the algebra \(\mathfrak T [R]\) in the case where \(R\) is a principal ideal domain. We are however especially interested precisely in the situation where this is not the case, i.e. where the number field \(K\) has nontrivial class group. To treat this case adequately we have to impose certain conditions on the range projections of the isometries \(s_a\). The most efficient way to formalize these conditions is to use projections associated with ideals in \(R\) as additional generators and to describe their relations. We mention that a description of the C*algebra generated by the left regular representation of a cancellative semigroup by analogous generators and relations had been discussed before also by Li, [15], Appendix A2, see also [20], Chapter 4 for a specific example.
An important role in our analysis of \(\mathfrak T [R]\) is played by a canonical maximal commutative subalgebra. Its Gelfand spectrum \(Y_R\) can be understood as a completion, for a natural metric, of the disjoint union \(\bigsqcup R/I\) over all nonzero ideals \(I\) of \(R\). It contains the profinite completion \(\hat{R}\) of \(R\) (which is the spectrum of the analogous commutative subalgebra of \(\mathfrak A [R]\)). It is important to note that the algebra \(\mathfrak T [R]\) is functorial for homomorphisms between rings while \(\mathfrak A [R]\) is not. This is reflected in the striking fact that the construction \(R\mapsto Y_R\) is contravariant under ring homomorphisms rather than covariant as one might expect. An inclusion of rings \(R\subset S\) induces a surjective map \(Y_S\rightarrow Y_R\). The same holds for the locally compact version of \(Y_R\) (corresponding to a natural stabilization of \(\mathfrak T [R]\)) which plays the role of the locally compact space of finite adeles.
Especially important for us is a natural oneparameter group \((\sigma _t)_{t\in \mathbb R }\) of automorphisms of \(\mathfrak T [R]\). It is closely related to Bost–Connes systems [1] and to Dedekind \(\zeta \)functions. In special cases it had been considered before in [4, 13].
The Toeplitz algebra for the semigroup \(\mathbb N \rtimes \mathbb N ^\times \)—which is very closely related to the Toeplitz algebra \(\mathfrak T [\mathbb Z ]\) for the ring \(\mathbb Z \) in the sense of the present paper—has been analyzed in [13]. In particular it was found in that paper that the canonical oneparameter automorphism group on this algebra has an intriguing KMSstructure. There is a phase transition at \(\beta =2\) with a spontaneous symmetry breaking. In the range \(1\le \beta \le 2\) there is a unique KMSstate while for \(\beta >2\) there is a family of KMSstates labeled by the probability measures on the circle and with partition function the Riemann \(\zeta \)function.
It turns out that, for our Toeplitz algebra, the KMSstructure is similar, but quite a bit more intricate. We show in Theorem 6.7 that for \(\beta \) in the range \(1\le \beta \le 2\) (with \(\beta =1\) playing a special role) there is a unique KMS state. The essential new feature which is also the source of the main technical difficulties in this paper is the presence of the class group, in the case where \(R\) is not a principal ideal domain. Our proof for the uniqueness of the KMSstate requires a delicate estimate of the asymptotics of partial Dedekind \(\zeta \)functions for different ideal classes, see Theorem 6.6. This theorem seems to be new and of independent interest. We include the proof in the appendix.
For \(\beta >2\) we obtain a splitting of the KMS states over the class group \(\Gamma \) for the number field \(K\). The KMS states for each \(\beta \) in this range are labeled by the elements \(\gamma \in \Gamma \), but moreover also by traces on a crossed product \(\mathcal C (\mathbb T ^n)\rtimes R^*\) (\(n\) being the degree of our field extension) by an action (which depends on \(\gamma \)) of the group \(R^*\) of units of \(R\). For a precise statement see Theorem 7.3. The partition functions are the partial Dedekind \(\zeta \)functions \(\zeta _\gamma \) associated with the ideal classes \(\gamma \) for \(K\).
In Sect. 8 we determine the ground states. We find a situation which is similar to the one for the KMS states in the range \(\beta >2\). The ground states are labeled by the states of a certain subalgebra of \(\mathfrak T [R]\).
We mention that our methods also immediately yield the KMSstructure of the much simpler, but in the case of a nontrivial class group still interesting, C*dynamical system that one obtains from the Toeplitz algebra of the multiplicative semigroup \(R^\times \) [(i.e. the C*algebra generated by the left regular action of this semigroup on \(\ell ^2(R^\times )\)] with the analogous oneparameter automorphism group, see Remark 7.5.
When we restrict to the case of a trivial class group, all our arguments become very simple indeed and can be used to get a simpler approach to the results in [13].
The presentation of \(\mathfrak T [R]\) in terms of generators and relations and the functoriality from Sect. 3 had been obtained and announced by the first named author before the present paper took shape. These two results have since been generalized to more general semigroups by Li [16]. The first named author is indebted to Peter Schneider for very helpful comments.
After this paper was circulated, Neshveyev [11] informed us that, using the crossed product description of \(\mathfrak T [R]\) in Sect. 5 and the methods developed, the KMSstructure on \((\mathfrak T [R], (\sigma _t))\) could be linked to that of a Bost–Connes system. The KMSstructure of this Bost–Connes system in turn was determined in [10]. Together, this would give a basis for an alternative approach to our results on KMSstates in Sects. 6 and 7.
We include a brief list of notations at the end of the appendix.
2 The Toeplitz algebra for the \(ax+b\)semigroup over \(R\)
Let \(R\) be the ring of algebraic integers in the number field \(K\). The \(ax+b\)semigroup for \(R\) is the semidirect product \(R\rtimes R^\times \) of the additive group \(R\) and the multiplicative semigroup \(R^\times = R\setminus \{0\}\) of \(R\). We can define the Toeplitz algebra for the semigroup \(R\rtimes R^\times \) as the C*algebra generated by the left regular representation of \(R\rtimes R^\times \) on \(\ell ^2(R\rtimes R^\times )\). We set out to describe this C*algebra abstractly as a C*algebra given by generators and relations.
Definition 2.1
 Ta:
The \(u^x\) are unitary and satisfy \(u^xu^y=u^{x+y}\), the \(s_a\) are isometries and satisfy \(s_as_b=s_{ab}\). Moreover we require the relation \(s_a u^x=u^{ax}s_a\) for all \(x\in R,\,a\in R^\times \).
 Tb:
The \(e_I\) are projections and satisfy \(e_{I\cap J}=e_Ie_J\), \(e_R=1\).
 Tc:
We have \(s_ae_Is_a^* = e_{aI}\).
 Td:
For \(x\in I\) one has \(u^xe_I = e_Iu^x\), for \(x\notin I\) one has \(e_Iu^xe_I=0\).
The first condition Ta simply means that the \(u^x\) and \(s_a\) define a representation of the semigroup \(R\rtimes R^\times \). We will see below that \(\mathfrak T [R]\) is actually isomorphic to the Toeplitz algebra for the \(ax+b\)semigroup \(R\rtimes R^\times \), see Corollary 4.16.
In the following, ideals in \(R\) will always be understood to be nonzero.
Remark 2.2
Remark 2.3
 (a)
The elements \(s_a\) and \(s_b^*\) commute if and only if \(a\) and \(b\) are relatively prime (i.e. \(aR+bR=R\)). (Proof: \(s_b^*s_a = s_as_b^*\) iff \(s_bs_b^*s_as_a^*= s_bs_as_b^*s_a^*\) iff \(e_{aR}e_{bR} = e_{abR}\). Then use the fact, established below using explicit representations of \(\mathfrak T [R]\), that \(e_I=e_J\Rightarrow I=J\))
 (b)
From condition Td it follows that \(e_Iu^xe_J=0\) if \(x\notin I+J\) and that \(e_Iu^xe_J=u^{x_1}e_{I\cap J}u^{x_2}\) if there are \(x_1\in I\) and \(x_2\in J\) such that \(x=x_1+x_2\).
Lemma 2.4
 (a)For any two ideals \(I\) and \(J\) in \(R\) one has$$\begin{aligned} e_If_J =\sum _{x\in I/(I\cap J)}u^xe_{I\cap J}u^{x}\quad f_If_J =\sum _{x\in (I+J)/(I\cap J)}u^xe_{I\cap J}u^{x} \end{aligned}$$
 (b)
If \(I\) and \(J\) are relatively prime, then \(f_If_J=f_{IJ}\). If \(I\subset J\), then \(f_If_J =f_I\).
 (c)
If \(I\) and \(J\) are relatively prime, then \(\varepsilon _I\varepsilon _J=\varepsilon _{IJ}\). If \(I\) and \(J\) have a common prime divisor but occuring with different multiplicities, then \(\varepsilon _I\varepsilon _J=0\).
 (d)
The family of projections \(\{e_I,f_I,\varepsilon _I \big \,I \,\text{ an} \text{ ideal} \text{ in} \,R\}\) is commutative.
 (e)
\(u^x f_Iu^{x}=f_I\) and \(u^x \varepsilon _Iu^{x}=\varepsilon _I\) for all \(x\in R\).
Proof
 (a)
Obvious from Remark 2.3.
 (b)
is a special case of the formula under (a).
 (c)
follows from the definition together with the fact that \(\varepsilon _{P^n}\varepsilon _{P^m}=0\) for a prime ideal \(P\) and \(n\ne m\).
 (d)
It follows from (a) that the \(e_I\) and \(f_I\) form a commutative family. However the \(\varepsilon _I\) are defined as products of differences of certain \(f_J\).
 (e)
follows directly from the definition.\(\square \)
3 Functoriality of \({\mathfrak T }[R]\) for injective homomorphisms of rings
We assume that we have an inclusion \(R\subset S\) of rings of algebraic integers. We are going to show that this induces an (injective) homomorphism \(\kappa : \mathfrak T [R]\rightarrow \mathfrak T [S]\). Denote by \(s_a,u^x,e_I\) the generators of \(\mathfrak T [R]\) and by \(\bar{s}_a,\bar{u}^x,\bar{e}_I\) the generators of \(\mathfrak T [S]\).
The homomorphism \(\kappa \) will map \(s_a\) to \(\bar{s}_a\), \(u^x\) to \( \bar{u}^x\) and it is clear that this respects the relations Ta. With an ideal \(I\) in \(R\) we associate the ideal \(IS\) in \(S\) and we define \(\kappa (e_I)=\bar{e}_{IS}\). It is then clear that relation Tc is also respected. The fact that Tb and Td are respected follows from the following elementary (and wellknown) lemma.
Lemma 3.1
 (a)
\(IS\cap R=I\)
 (b)
\(IS\cap JS=(I\cap J)S\)
Proof
Both statements can be proved in an elementary way using the unique decomposition of \(I\) and \(J\) into prime ideals in \(R\), cf. [19], p. 45 and p. 52, Exercise 1. The statements also follow from the fact that \(S\) is a flat (even projective) module over \(R\), see [2], Chap. I §2.6 Prop. 6 and Corollary.\(\square \)
Summarizing, we obtain
Proposition 3.2
Let \(R\) and \(S\) be the rings of algebraic integers in the number fields \(K\) and \(L\), respectively. Then any injective homomorphism \(\alpha :R\rightarrow S\) induces naturally a homomorphism \(\mathfrak T [R]\rightarrow \mathfrak T [S]\).
It follows from Theorem 4.13 below that this homomorphism is also injective.
4 The canonical commutative subalgebra
Lemma 4.1
 (a)
If \(d\in \bar{\mathcal D }\), then \(s_ads_a^*\) and \(u^xdu^{x}\) are in \(\bar{\mathcal D }\) for all \(a\in R^\times ,\,x\in R\).
 (b)
The set of linear combinations of elements of the form \(s_a^* du^xs_b\) with \(a,b\in R^\times ,\;x\in R,\,d\in \bar{\mathcal D }\) is a dense \(\star \)subalgebra in \(\mathfrak T [R]\).
Proof
 (a)
This follows from the definition and conditions Ta–Td.
 (b)
The set of elements of the form \(s_a^* du^xs_b\) contains the generators and, by (a), is invariant under adjoints and multiplication from the left or from the right by elements \(s_c,s_c^*, u^y, e_I\) for \(c\in R^\times ,\,y\in R\), \(I\) an ideal in \(R\) (the invariance under multiplication by \(s_c^*\) on the right follows from the identity \(s_bs^*_c=s^*_cs_cs_bs^*_c=s^*_cs_bs_cs^*_c=s^*_ce_{bcR}s_b\)).\(\square \)
Lemma 4.2
Proof
Lemma 4.3
The algebra \(\varepsilon _{P^n}\bar{\mathcal D }_P\) is finitedimensional and isomorphic to \(\mathcal C (R/P^{n1})\). For each \(\{x\}\) in \(\mathcal C (R/P^{n1})\), the projection \(\,\delta _{P^{n1}}^x = u^xe_{P^{n1}}\varepsilon _{P^n}u^{x},\,x\in R/P^{n1}\) is minimal in this algebra and corresponds to the characteristic function of \(\{x\}\). The isomorphism \(\varepsilon _{P^n}\bar{\mathcal D }_P\cong \mathcal C (R/P^{n1})\) is compatible with the natural action of the additive group \(R\) on these two algebras.
Proof
For \(k\ge n\), since \(e_{P^k}\le f_{P^n}\), we have \(e_{P^k}\varepsilon _{P^n}=0\) and, since \(u^x\) commutes with \(\varepsilon _{P^n}\), also \(u^xe_{P^k}u^{x}\varepsilon _{P^n}=0\) for such \(k\).
On the other hand if \(k\le n1\), then \(e_{P^k}u^xe_{P^{n1}}=0\) for \(x\notin P^k\) and \(e_{P^k}u^xe_{P^{n1}}=u^xe_{P^{n1}}\) for \(x\in P^k\). Applying this to the product \(e_{P^k}\delta ^x_{P^n}=e_{P^k}u^x e_{P^n}u^{x}\varepsilon _{P^{n+1}}\) we see that this expression vanishes for \(x\notin P^k\) and equals \(\delta ^x_{P^n}\) for \(x\in P^k\).
The last assertion then is an immediate consequence.\(\square \)
Denote by \(\mathcal D _P\) the ideal in \(\bar{\mathcal D }_P\) generated by the \(\varepsilon _{P^n}\). Lemma 4.3 shows that \(\mathcal D _P\cong \bigoplus \mathcal C (R/P^n)\). Since the union of the subalgebras \(G_k\) is dense in \(\bar{\mathcal D }_P\), the last statement in this lemma also shows that \(\mathcal D _P\) is an essential ideal in \(\bar{\mathcal D }_P\).
Let \(\iota : \mathcal C (R/P^{n})\rightarrow \mathcal C (R/P^{m})\) denote the homomorphism induced by the quotient map \(R/P^m\rightarrow R/P^{n}\) for \(m>n\).
Lemma 4.4
Proof
Lemma 4.5
Proof
Now let \(P_1, P_2, \ldots \) be an enumeration of the prime ideals in \(R\) (say ordered by increasing norm \(R/P_i\)) and, for each \(n\), let \(\mathcal I _n\) be the set of ideals of the form \(I =P_1^{k_1}P_2^{k_2}\cdots P_n^{k_n}\) with all \(k_i \ge 0\). We write \(\mathcal D _n = \mathcal D _{P_1}\mathcal D _{P_2}\cdots \mathcal D _{P_n}\) and \(\bar{\mathcal D }_n = \bar{\mathcal D }_{P_1}\bar{\mathcal D }_{P_2}\cdots \bar{\mathcal D }_{P_n}\). The \(\bar{\mathcal D }_{P_n}\) all commute and \(\bar{\mathcal D }\) obviously is the inductive limit of the \(\bar{\mathcal D }_n\).
The unitaries \(u^x\), \(x\in R\) act componentwise on \(\ell ^2(R/I)\) in the natural way.
The isometries \(s_a\) act through the composition: \(\ell ^2(R/I)\cong \ell ^2 (aR/aI) \hookrightarrow \ell ^2 (R/aI)\).
The projection \(e_J\) is represented by the orthogonal projection onto the subspace \( H = \bigoplus _{I\subset J} \ell ^2 (J/I) \) of \(H\).
Lemma 4.6
Let \(I=P_1^{k_1}P_2^{k_2}\cdots P_n^{k_n}\) with all \(k_i\ge 1\), and \(x_1,x_2,\ldots , x_n \in R\). Then \(\mu (\delta ^0_{P_1^{k_1}})\) acts on the subspace \(\ell ^2(R/I)\) of \(H_R\) as the orthogonal projection onto the subspace \(\ell ^2(P_1^{k_1}/I)\). Thus \(\mu (\delta ^0_{P_1^{k_1}}\delta ^0_{P_2^{k_2}}\cdots \delta ^0_{P_n^{k_n}})\) acts on this subspace as the orthogonal projection onto the onedimensional subspace \(\ell ^2(I/I)\) and \(\mu ( \delta ^{x_1}_{P_1^{k_1}}\delta ^{x_2}_{P_2^{k_2}}\cdots \delta ^{x_n}_{P_n^{k_n}})\) acts as the orthogonal projection onto the onedimensional subspace \(\ell ^2((I+z)/I)\) where \(z\) is the unique element in \(\bigcap _i (P_i^{k_i}+x_i)/I\).
Proof
This follows from the definition of \(\mu (e_{P_1^{k_1}})\) and the fact that \(\varepsilon _{P_1^{k_1+1}}=1\) on \(\ell ^2(R/I)\) (recall that, by definition \(\delta ^0_{P_1^{k_1}}=e_{P_1^{k_1}}\varepsilon _{P_1^{k_1+1}}\)).\(\square \)
Lemma 4.7
Proof
The identity \(\delta _{I,n}^0=e_I \varepsilon _{IP_1P_2\cdots P_n}\) follows from the equations \(e_I=e_{P_1^{k_1}}e_{P_2^{k_2}}\cdots e_{P_n^{k_n}}\) and \(\varepsilon _{IP_1P_2\cdots P_n}=\varepsilon _{P_1^{k_1+1}}\varepsilon _{P_2^{k_2+1}}\cdots \varepsilon _{P_n^{k_n+1}}\) (see Lemma 2.4). The second identity follows from the first one in combination with the corresponding identity in Lemma 4.2.\(\square \)
We have now shown that \(\mathcal D _n=\mathcal D _{P_1}\mathcal D _{P_2}\cdots \mathcal D _{P_n}\) is isomorphic to the tensor product \(\bigotimes _{1\le i\le n}\mathcal D _{P_i}\) with minimal projections the \(\delta _{I,n}^x\), \(I\in \mathcal I _n\). Thus \(\mathcal D _n\cong \bigoplus _{I\in \mathcal I _n}\mathcal C (R/I)\) and the spectrum of \(\mathcal D _n\) is \(\bigsqcup _{I\in \mathcal I _n} R/I\) (this is the cartesian product of the spectra \(\bigsqcup _{k\ge 0} R/P_i^k\) of the \(\mathcal D _{P_i}\)).
Corollary 4.8
\(\mathcal D _n\) is an essential ideal in \(\bar{\mathcal D }_n\). \(\bar{\mathcal D }_n\) is isomorphic to \(\bar{\mathcal D }_{P_1}\otimes \bar{\mathcal D }_{P_2}\cdots \otimes \bar{\mathcal D }_{P_n}\) and \(\bar{\mathcal D }\) is isomorphic to the infinite tensor product \( \bigotimes _P \bar{\mathcal D }_P \).
Proof
From Lemma 4.5 it follows that, for each prime ideal \(P\), we have \(\mathrm Spec \,\bar{\mathcal D }_P = \mathrm Spec \,\mathcal D _P\cup \mathrm Spec \,\mathcal C = \bigsqcup _n R/P^n\sqcup R_P\).
For an ideal \(I\) in \(R\) and \(x\in R/I\), we consider the projection \(e^x_I=u^xe_Iu^{x}\). Since, by Remark 2.3(b), \(e^x_Ie^y_J\) is either zero or equal to \(e^z_{I\cap J}\) for \(z\in (x+I)\cap (y+J)\), the set of projections \(\{e^x_I\,  \,\, I\,\text{ ideal} \text{ in} R,\,x\in R/I\}\) is multiplicatively closed.
Now, since \(\bar{\mathcal D } \cong \bigotimes _P \bar{\mathcal D }_P\), every character \(\varphi \) of \(\bar{\mathcal D }\) is of the form \(\varphi =\bigotimes _P\varphi _P\) with each \(\varphi _P\) either of the form \(\eta _{P^n}^x\) for \(n\in \mathbb N ,\,x\in R/P^n\) or \(\eta _z\) with \(z\in R_P\).
Proposition 4.9
The subset \(\bigsqcup _IR/I\) is dense in \(\mathrm Spec \,\bar{\mathcal D }\). Thus \(\mathrm Spec \,\bar{\mathcal D }\) is the completion of \(\bigsqcup _IR/I\) for the metric \(d_\alpha \) described above.
Proof
It is clear from the discussion above that the set of elements of the form \(\bigotimes _i\eta ^{x_i}_{P_i^{k_i}}\) is dense. We show that each element \(\eta =\bigotimes _i\eta ^{x_i}_{P_i^{k_i}}\) can be approximated by the \(\eta _I^x\). In fact, if \(I=P_1^{k_1}P_2^{k_2}\cdots P_n^{k_n}\) and \(x\in R\) is such that \(x/P_i^{k_i}=x_i,\, i=1,\ldots n\), then \(\eta (e^y_J)=\eta _I^x (e^y_J)\) for each ideal \(J\) that contains only \(P_1,\ldots , P_n\) in its prime ideal decomposition and for any \(y\in R/J\).\(\square \)
Remark 4.10
This description of \(\mathrm Spec \,\bar{\mathcal D }\) also clarifies the wrongway functoriality in \(R\) of the construction. Assume that we have field extensions \(\mathbb Q \subset K\subset L\) and corresponding inclusions \(\mathbb Z \subset R\subset S\) of the rings of algebraic integers. Denote by \(Y_R\) and \(Y_S\) the spectra of the corresponding commutative subalgebras \(\bar{\mathcal D }_R\) and \(\bar{\mathcal D }_S\) in \(\mathfrak T [R]\) and \(\mathfrak T [S]\), respectively. Thus \(Y_R\) and \(Y_S\) are completions of the metric spaces \(\bigsqcup _I R/I\) and \(\bigsqcup _J S/J\), respectively.
With every character \(\eta _J^x\in \bigsqcup _J S/J\) we can associate a character \((\eta _J^x)^{\prime }\in \mathrm Spec \,\bar{\mathcal D }_R\) by defining \((\eta _J^x)^{\prime } (e_M^y)= \eta _{J}^{x} (e_{MS}^y)\) for an ideal \(M\) in \(R\) and \(y\in R/M\). The map \(\eta _{J}^{x}\rightarrow (\eta _{J}^{x})^{\prime }\) is obviously contractive (up to a constant \(n^\alpha \) with \(n=[L:K]\)) for the metrics \(d_\alpha \) and thus extends to a continuous map \(\mathrm Spec \,\bar{\mathcal D }_S\rightarrow \mathrm Spec \,\bar{\mathcal D }_R\). It is surjective, since the dense subset \(\bigsqcup _I R/I\) of \(\mathrm Spec \,\bar{\mathcal D }_R\) has a natural lift to \(\mathrm Spec \,\bar{\mathcal D }_S\). In fact, one immediately checks that \((\eta _{IS}^x)^{\prime }=\eta _I^x\) for an ideal \(I\subset R\) and \(x\in R/I\).
Lemma 4.11
Proof
Composing these two expectations we obtain the faithful conditional expectation \(E: \mathfrak T [R]\rightarrow \bar{\mathcal D }\) which we will use now. Note that, for a typical element \(z=s_a^*du^xs_b\), \(E(z)=0\), unless \(a=b, x=0\) in which case \(E(z)=s_a^*ds_a\).
Lemma 4.12
 (a)
There is a minimal projection \(\delta \) in \(\mathcal D _n\) such that \(\Vert d\delta \Vert =\Vert \delta d\delta \Vert \ge \Vert d\Vert \varepsilon \).
 (b)
There is \(k>0\) and a minimal projection \(\delta ^{\prime }\in \mathcal D _{n+k}\), \(\delta ^{\prime }\le \delta \) such that \(\delta ^{\prime }s^*_{a_i}d_iu^{x_i}s_{b_i}\delta ^{\prime }=0\) for all \(i\).
 (c)
For the projection \(\delta ^{\prime }\) in \((\)b\()\) one has \(\Vert \delta ^{\prime } z\delta ^{\prime }\Vert \ge \Vert E(z)\Vert \varepsilon \)
Proof
 (a)
simply expresses the fact that \(\mathcal D _n\) is essential.
 (b)Let \(\delta =\delta _{I,n}^y\), \(I\in \mathcal I _n\), \(y\in R/I\). Using Lemma 4.11 we may then choosesuch that for each \(i\) the projections \(s_{a_i}\delta ^{\prime }s^*_{a_i}\) and \(u^{x_i}s_{b_i}\delta ^{\prime }s^*_{b_i}u^{x_i}\) are orthogonal. This projection \(\delta ^{\prime }\) will have the required properties.$$\begin{aligned} \delta ^{\prime }=u^y \delta _{I,n}^0\delta ^0_{P_{n+1}^{t_1}}\cdots \delta ^0_{P_{n+k}^{t_k}}u^{y} \end{aligned}$$
 (c)
follows immediately from (a) together with (b) using the fact that \(E(z)=d\) and that \(d\delta \) is just a multiple of \(\delta \) (\(\delta \) is a minimal projection in \(\mathcal D _n\)).\(\square \)
Theorem 4.13
 (a)
\(\alpha \) is injective.
 (b)
\(\alpha \) is injective on \(\bar{\mathcal D }\).
 (c)
\(\alpha \) is injective on \(\mathcal D _n\) for each \(n\).
Proof
(c) implies (b) since \(\mathcal D _n\) is essential in \(\bar{\mathcal D }_n\) for each \(n\).
(b) \(\Rightarrow \) (a): Let \(h\) be a positive element in \(\mathfrak T [R]\) with \(\alpha (h)=0\) and \(z\) a linear combination as in 4.12 such that \(\Vert hz\Vert <\varepsilon \). Let \(\delta ^{\prime }\) be a projection as in 4.12 (b) such that \(\Vert \delta ^{\prime } z\delta ^{\prime }\Vert \ge \Vert E(z)\Vert \varepsilon \) (and such that \(\delta ^{\prime }z\delta ^{\prime }\) is a multiple of \(\delta ^{\prime }\)). If \(\alpha (h)=0\), then \(\Vert \alpha (\delta ^{\prime }z\delta ^{\prime })\Vert <\varepsilon \) and thus also \(\Vert \delta ^{\prime }z\delta ^{\prime }\Vert <\varepsilon \). It follows that \(\Vert E(h)\Vert <2\varepsilon \). Since this holds for each \(\varepsilon \), \(E(h)=0\) and, since \(E\) is faithful, \(h=0\).\(\square \)
From this technical theorem we can derive the following important Corollaries 4.14,4.16 and 4.17.
Corollary 4.14
The representation \(\mu \) of \(\mathfrak T [R]\) on \(\bigoplus _I \ell ^2(R/I)\) is an isomorphism.
Proof
The restriction of \(\mu \) to each \(\mathcal D _n\) is injective by Lemma 4.6.\(\square \)
Let \(\mathfrak T \subset \mathcal L (\ell ^2(R\rtimes R^\times ))\) denote the C*algebra defined by the left regular representation of the semigroup \(R\rtimes R^\times \) (cf. Sect. 2). Given an ideal \(I\) in \(R\) we can define a projection \(e^{\prime }_I\) in \(\mathcal L (\ell ^2(R\rtimes R^\times ))\) as the orthogonal projection on the subspace \(\ell ^2(I\rtimes I^\times )\subset \ell ^2(R\rtimes R^\times )\). Denote by \(u^{\prime x}\) and \(s^{\prime }_a\) the operators defined by the left action of \(R\) and \(R^\times \) on \(\ell ^2(R\rtimes R^\times )\). Then it is easy to check that the \(u^{\prime x}\), \(s^{\prime }_a\) and \(e^{\prime }_I\) satisfy the relations defining \(\mathfrak T [R]\).
Lemma 4.15
 (a)
Every ideal \(I\) in \(R\) can be written in the form \(\frac{a}{b}R\cap R\) with \(a,b\in R^\times \).
 (b)
If \(I =\frac{a}{b}R\cap R\), then \(e_I =s_b^*s_as_a^*s_b\) and, similarly, \(e^{\prime }_I = s_b^{^{\prime }*} s_a^{\prime } s_a^{^{\prime }*} s_b^{\prime }\).
Proof
 (a)
Let \(Q\) and \(M\) be ideals such that \(I,Q,M\) are relatively prime and such that \(IQ\), \(QM\) are principal, say \(IQ=aR\), \(QM=bR\). Then \(bI=IQM=IQ\cap QM= aR\cap bR\).
 (b)
One has \(bI = aR\cap bR\) and therefore \(s_b e_Is_b^*=s_as_a^*s_bs_b^*\). Since this uses only the relations defining \(\mathfrak T [R]\), it also holds in \(\mathfrak T \).\(\square \)
This lemma shows that \(e^{\prime }_I\in \mathfrak T \), hence we obtain a natural homomorphism \(\mathfrak T [R]\rightarrow \mathfrak T \) by assigning \(s_a \mapsto s^{\prime }_a,\,u^x\mapsto u^{\prime x},\, e_I\mapsto e^{\prime }_I\).
Corollary 4.16
The natural map \(\mathfrak T [R]\rightarrow \mathfrak T \) is an isomorphism.
Proof
The map is obviously surjective. To prove injectivity, suppose \(I \in \mathcal I _n\) and \(x \in R\) are given. Let \(Q\not \in \{P_1, P_2,\ldots ,P_n\}\) be a prime ideal in the ideal class \([I]^{1}\) and let \(a \) be a generator of the principal ideal \(IQ\) (for the existence of such a \(Q\) see for instance [17], chapter 7, §2, Corollary 7). Since the exponents of \(\{P_1, P_2,\ldots ,P_n\}\) in the prime factorization of \(aR\) are identical to those of \(I\), the image of the projection \(\delta _{I,n}^x\) fixes the canonical basis vector \(\xi _{(x,a)}\in \ell ^2(R\rtimes R^\times )\), so it does not vanish. Hence the natural map is injective on \(\mathcal D _n\) for each \(n\) and therefore injective by Theorem 4.13.\(\square \)
Denote, as above, by \(Y_R\) the spectrum of \(\bar{\mathcal D }\). The semigroup \(R\rtimes R^\times \) acts on \(Y_R\) in a natural way and this action corresponds to the canonical action of \(R\rtimes R^\times \) on \(\bar{\mathcal D }\) by conjugation by the \(u^x\) and the \(s_a\). We use the definition of a semigroup crossed product as in [12], Sect. 2, [15], Appendix A1.
Corollary 4.17
The algebra \(\mathfrak T [R]\) is canonically isomorphic to the semigroup crossed product \((\mathcal C (Y_R)\rtimes R)\rtimes R^\times \).
Proof
The generators \(e_I^0\) of \(\mathcal C (Y_R)\) together with the canonical generators of \(R\rtimes R^\times \) in \((\mathcal C (Y_R)\rtimes R)\rtimes R^\times \) satisfy the relations defining \(\mathfrak T [R]\), hence they determine a surjective homomorphism \(\mathfrak T [R]\rightarrow (\mathcal C (Y_R)\rtimes R)\rtimes R^\times \). This homomorphism is injective on \(\bar{\mathcal D }\) and therefore injective by 4.13.\(\square \)
5 An alternative description of \(\mathrm Spec \,\bar{\mathcal D }\) and the dilation of \(\mathfrak T [R]\) to a crossed product by \(K\rtimes K^*\)
We will give a parametrization of the spectrum of \(\bar{\mathcal D }\), along the lines of that obtained for the case \(R= \mathbb Z \) in [9, 13], and use it to realize \(\mathfrak T [R]\) as a full corner in a crossed product. Let \({\mathbb A _f}\) denote the ring of finite adeles over \(K\) and let \({\hat{R}}\) be the compact open subring of (finite) integral adeles; their multiplicative groups are the finite ideles \({\mathbb A ^*_f}\) and the integral ideles \({\hat{R}}^*\), respectively.
When \(a\) and \(b\) are superideals such that \(\epsilon _P(a) \le \epsilon _P(b)\) for every \(P\) we write \(a\le b\). In this case \(b{\hat{R}}\subset a{\hat{R}}\) and there is an obvious homomorphism reduction modulo\(a\) of \({\mathbb A _f}/b{\hat{R}}\) to \( {\mathbb A _f}/a{\hat{R}}\); we will write \(r(a)\) for the image of \(r\in {\mathbb A _f}/b{\hat{R}}\). When \(I\) and \(J\) are ideals of \(R\) viewed as elements of \({\mathbb A _f}/{\hat{R}}^*\), then \(I\le J\) means \(J \subset I\) and the reduction defined above is the usual reduction of ideal classes \(R/J \rightarrow R/I\).
Proposition 5.1
Proof
Clearly \((R^{\times })^{1}({R \rtimes R^\times }) = {K\rtimes K^*}\), so \({R \rtimes R^\times }\) is an Ore semigroup, and \( \cup _{k\in R^{\times }} (0,k)^{1}\Omega _{\hat{R}}\) is dense in \(\Omega _{\mathbb A _f}\) because for every element \(\omega _{r,a} \in \Omega _{\mathbb A _f}\) there exist \(k\in R^{\times }\) such that \(kr \in {\hat{R}}\) and \(ka \in {\hat{R}}/ ka{\hat{R}}\). By Laca [8, Theorem 2.1] the action of \({K\rtimes K^*}\) on \(\mathcal C _0(\Omega _{\mathbb A _f})\) is the minimal automorphic dilation (see [8]) of the semigroup action of \({R \rtimes R^\times }\) on \(\mathcal C (\Omega _{\hat{R}})\). The fullness of \({{\mathchoice{1\!\!1}{1\!\!1}{1\!\!1}{1\!\!1}}}_{\Omega _{\hat{R}}}\) and the isomorphism to the corner then follow by Laca [8, Theorem 2.4].\(\square \)
Proposition 5.2
Let \(v_{m,k}\) with \((m,k)\in {R \rtimes R^\times }\) be the semigroup of isometries in \(\mathcal C (\Omega _{\hat{R}}) \rtimes {R \rtimes R^\times }\) implementing the action of \(R\rtimes R^\times \). For each ideal \(I\) in \(R\) let \(E_I\) be the characteristic function of the set \(\{\omega _{s,b} \in \Omega _{\hat{R}}\,\, b \ge I, \, s(b) \in I\}\). Then the maps \(u^x \mapsto v_{x,1}\), \(s_k \mapsto v_{0,k}\), and \(e_I \mapsto E_I\) extend to an isomorphism of \(\mathfrak T [R]\) onto \(\mathcal C (\Omega _{\hat{R}}) \rtimes {R \rtimes R^\times }\).
Proof
The set \(\{\omega _{s,b} \in \Omega _{\hat{R}}: b \ge I, \, s(b) \in I\}\) is closed open because it is defined via finitely many conditions [(\(\epsilon _P(b) \ge \epsilon _P(I)\) on the prime factors of \(I\) and \(s=0 \text{ mod} I\)] each of which determines a closed open set; thus \(E_I\) is continuous. The relations Ta are satisfied because \((m,k) \rightarrow v_{m,k}\) is an isometric representation of \({R \rtimes R^\times }\), and Tb holds because \(b \ge I \) and \(b \ge J\) if and only if \(b \ge I \cap J\), and \(s(I) \in I \) and \( s(J) \in J\) if and only if \(s(I\cap J) \in I \cap J\). Computing with \(m=0\) in Eq. (1) shows that multiplication by \(k\in R^{\times }\) maps the support of \(E_I\) onto the support of \(E_{kI}\), hence relation Tc holds. Similarly, setting \(k=1\) in Eq. (1) shows that addition of \(m\) maps the support of \(E_I\) onto itself if \(m\in I\), and onto a set disjoint from it if \(m\not \in I\), showing that relation Td holds. This gives a homomorphism \(h: \mathfrak T [R] \rightarrow \mathcal C (\Omega _{\hat{R}})\rtimes {R \rtimes R^\times }\).
To show that \(h\) is surjective it suffices to prove that the functions \(E_I^x:= v_{x,1}E_I v_{x,1}\) separate points in \(\Omega _{\hat{R}}\). So let \( \omega _{r,a}\) and \(\omega _{s,b}\) be two distinct points in \(\Omega _{\hat{R}}\). If \(a \ne b\), we may assume there exists a prime ideal \(Q\) such that \(\epsilon _Q(a) < \epsilon _Q(b)\) (otherwise reverse the roles of \(a\) and \(b\)). If we now let \(I = Q^{\epsilon _Q(b)}\), then \(E_I^{s(I)}\) takes on the value \(1\) at \(\omega _{s,b}\) but vanishes at \(\omega _{r,a}\). If \(a = b\) as superideals, since the points \(\omega _{r,a}\) and \(\omega _{s,b}\) are distinct, there exists an ideal \(I \le a\) for which \(r(I) \ne s(I)\), in which case the function \(E_I^{s(I)}\) does the separation.
Next we show that this homomorphism is injective on \(\mathcal D _n\) for each \(n\). Fix \(n\), let \(I\) be an ideal whose prime factors are all in \(\{P_1, P_2, \ldots , P_n\}\) and choose a class \(x\in R/I\). Choose \(a\in I\) such that \(\varepsilon _{P_j}(a) = \varepsilon _{P_j}(I)\) for \(j = 1, 2. \ldots , n\) (if \(I\) is principal, a generator will do; otherwise adjust with a prime ideal \(Q \notin \{P_1, P_2, \ldots , P_n\}\) such that \(IQ\) is principal). Also choose \(r\in {\hat{R}}/a{\hat{R}}\) such that \(r(I) = x\) in \(R/I\). Then \(E_I^x (\omega _{r,a}) = 1\), but \(E_{IPj}^x (\omega _{r,a}) = 0\) for each \(j\), proving that \(h(\delta _{I,n}^x ) \ne 0\). Hence \(h\) is injective on \(\mathcal D _n\) and the result follows by Theorem 4.13.\(\square \)
As a byproduct we see that \(\Omega _{\hat{R}}\) is an ‘adelic’ realization of the spectrum of \(\bar{\mathcal D }\).
Corollary 5.3
We view each nonzero ideal \(I\) of \(R\) as an element \(a(I)\) of \({\hat{R}}/{\hat{R}}^*\) and, similarly, we view each \(x\in R/I\) as a class \(r(x,I)\) in \({\hat{R}}/a(I) {\hat{R}}\cong R/I\). Then the map \(\eta _I^x \mapsto \omega _{r(x,I),a(I)}\) defined for \((x,I) \in \bigsqcup _I R/I\) extends to a homeomorphism of the spectrum \(Y_R\) of \(\bar{\mathcal D }\) onto \(\Omega _{\hat{R}}\).
Proof
6 KMSstates for \(\beta \le 2\)
Recall that for a nonzero ideal \(I\) in \(R\) we denote by \(N(I)\) the norm of \(I\), i.e. the number \(N(I)=R/I\) of elements in \(R/I\). For \(a\in R^\times \) we also write \(N(a)=N(aR)\). The norm is multiplicative, [19].
Proposition 6.1
There are no \(\beta \)KMS states on \(\mathfrak T [R]\) for \(\beta <1\).
Proof
Lemma 6.2
Let \(\varphi \) be a \(\beta \)KMS state for \(\beta > 1\) on \(\mathfrak T [R]\) and let \(\pi _\varphi \) be the associated GNSrepresentation on \(H_\varphi \). Then \(\pi _\varphi (\mathcal D _n)H_\varphi \) is dense in \(H_\varphi \), for each \(n\).
Proof
Fix \(n\in \mathbb N \) and let \(J\) be an ideal in \(\mathcal I _n\). Since the class group for the field of fractions \(K\) is finite, there is \(k\in \mathbb N \) such that \(J^k\) is a principal ideal, say \(J^k=aR\) with \(a\in R^\times \). We have \(N(J^k) =N(a)\).
The subspace \(L=\overline{\pi _\varphi (\mathcal D _n)H_\varphi }\) is invariant under all \(u^x\), \(x\in R\). It is also invariant under all \(s_c, s_c^*\), \(c\in R^\times \). The reason is that if \(cR=QS\) with \(S \in \mathcal I _n\) and \(Q\) relatively prime to \(P_1, P_2, \ldots , P_n\), then, according to Lemma 4.11, for every \(I\in \mathcal I _n\) we have \(s_c\delta ^x_{I}s^*_c = \delta ^{cx}_{SI} (u^{cx}e_Q u^{cx})\le \delta ^{cx}_{SI}\,\in \mathcal D _n\).
Denote by \(E\) the orthogonal projection onto \(L^\perp \). Then \(1E\) is the strong limit of \(\pi _\varphi (h^{1/n})\) where \(h\) is a strictly positive element in \(\mathcal D _n\). Therefore \(\varphi _E\) defined by \(\varphi _E (z)=(E\pi _\varphi (z)\xi _\varphi \xi _\varphi )\), for the cyclic vector \(\xi _\varphi \) in the GNSconstruction, is a \(\beta \)KMS functional (consider the limit \(n\rightarrow \infty \) of the expression \(\varphi ((1h^{1/n})x\sigma _{i\beta }(y))=\varphi (y(1h^{1/n})x)\) using the fact that \(E\) commutes with \(y\)). Consider the restriction \(\rho \) of \(\pi _\varphi \) to \(L^\perp \). Then \(\rho (\mathcal D _n)=0\), whence \(\rho (1f_{J^k})=0\).
Lemma 6.3
Let \(\varphi \) be a \(\beta \)KMS state for \(1\le \beta \le 2\) on \(\mathfrak T [R]\) and let \(I\) be a fixed ideal in \(R\). Then \(\varphi (\delta ^0_{I,n})\) tends to 0 for \(n \rightarrow \infty \).
Proof
Lemma 6.4
Proof
Lemma 6.5
Let \(\varphi \) be a \(\beta \)KMS state for \(1\le \beta \le 2\) on \(\mathfrak T [R]\). Let \(\bar{\mathcal D }\) be the canonical subalgebra of \(\mathfrak T [R]\) generated by all projections \(u^xe_Iu^{x}\) and let \(d\in \bar{\mathcal D }\). Then \(\varphi (s_a^*du^ys_b)\) is zero except if \(a=b\) and \(y=0\) (in which case the argument \(s_a^*du^ys_b\) is also an element in \(\bar{\mathcal D }\)).
Proof
Assume first that \(g=1\). Then \(z_I^x\ne 0\) only if \(bx \equiv bx+y \mod \,aI\), that is, only if \(y\in bI\). For a fixed \(y\ne 0\) this last condition is satisfied only for the finitely many ideals \(I\) in \(R\) such that \(bI\) divides \(yR\). Thus, if \(y\ne 0\), in the sum (5) there are at most a fixed finite number (independent of \(n\)) of nonzero terms and each individual term is bounded by \(\varphi (\delta _{I,n}^0) \Vert z\Vert \), which is arbitrarily small for large \(n\) by Lemma 6.3, whence \(\varphi (z)=0\).
As a consequence of this lemma, in order to know \(\varphi \), it suffices to know its values on \(\bar{\mathcal D }\). Moreover, for \(1< \beta \le 2\) it suffices to know \(\varphi \) on \(\mathcal D _n\) for all \(n\), by Lemma 6.2.
Theorem 6.6
We postpone the proof and give it in the appendix.
Theorem 6.7
Proof
Suppose \(\varphi \) is a \(\beta \)KMS state. Lemma 6.5 implies that \(\varphi \) factors through the conditional expectation \(E:\mathfrak T [R] \rightarrow \bar{\mathcal D }\) for \(1\le \beta \le 2\).
The next step is to show that (6) holds. Since the linear combinations of the projections \(e_I^x:=u^xe_Iu^{x}\) are dense in \(\bar{\mathcal D }\) and since \(\varphi (u^xe_Iu^{x})=\varphi (e_I)\), this will yield the uniqueness assertion. The argument for \(\beta =1\) is easier and we do it first.
It is obvious that such a state \(\varphi \) satisfies \(\varphi (f_P) =1\) and hence vanishes on the projections of the form \(\varepsilon _P\) that generate the kernel of the quotient map \(q: \mathfrak T [R]\rightarrow \mathfrak A [R]\) as an ideal, so \(\varphi \) factors through this quotient. It is now easy to prove existence of a \(1\)KMS state. From Sect. 4 of [5], and the fact that \(q\) intertwines the canonical conditional expectations on \(\mathfrak T [R]\) and on \(\mathfrak A [R]\), we know that the image of \(\bar{\mathcal D }\) in \(\mathfrak A [R]\) under \(q\) is naturally isomorphic to \(C(\hat{R})\) (\(\hat{R}\) being the profinite completion of \(R\)). If we let \(\lambda _1\) be the state of \(C(\hat{R})\) given by normalized Haar measure on \(\hat{R}\), an easy computation shows that \(\varphi _1:= \lambda _1 \circ E \circ q\) satisfies the \(1\)KMS condition from (2). This finishes the proof in the case \(\beta =1\).
Let us now prove existence in this case. Since \(\mathcal D _n\) is essential in \(\bar{\mathcal D }_n\) there is a natural embedding \(\bar{\mathcal D }_n\hookrightarrow \ell ^\infty (\mathrm Spec \,\mathcal D _n)\) and we know that \(\mathrm Spec \,\mathcal D _n=\bigsqcup _{I\in \mathcal I _n}R/I\). The minimal projections in \(\mathcal D _n\) are the \(\delta _{I,n}^x,\,I\in \mathcal I _n,\,x\in R/I\). Thus any \(d\) in \(\bar{\mathcal D }_n\) is represented by an \(\ell ^\infty \)function \((I,x) \mapsto \lambda _I^x( d)\) uniquely defined by \(d \delta _{I,n}^x = \lambda _I^x( d) \delta _{I,n}^x\) on \(\mathrm Spec \,\mathcal D _n\). Notice that for \(d\in \mathcal D _n\) one actually has \(d=\sum \lambda _I^x(d) \delta _{I,n}^x\).
Remark 6.8
7 KMSstates for \(\beta > 2\)
To obtain the most general KMSstate, we have to consider a more general family of representations of \(\mathfrak T [R]\). We fix temporarily a class \(\gamma \) in the class group \(\Gamma \) and we choose a reference ideal \(J=J_\gamma \) in this class.
Let \(\tau \) be a tracial state on the C*algebra \(C^*(J \rtimes R^*)\), where the semidirect product is taken with respect to the multiplicative action of the group of invertible elements (units) \(R^*\) on the additive group \(J\). Note that these traces form a Choquet simplex [21]. By Neshveyev [18, Corollary 5] the extreme points can be parametrized by pairs in which the first component is an ergodic \(R^*\)invariant probability measure \(\mu \) on the compact dual group \(\hat{J}\) on which the isotropy is a constant group \(\mu \)a.e., and the second component is a character of that isotropy group.
Lemma 7.1
Proof
To simplify the notation let \(U^x:= \pi _\tau (u^x)\) for \(x\in R\) and \(S_g:= \pi _\tau (s_g)\) for \(g\in R^*\). We may view the cyclic vector \(\xi _I \in H_I\) as a vector in \(\ell ^2 (R/I,H_I)\) (supported on the trivial class) which is cyclic for the action of \(C^*(R\rtimes R^*)\) on \(\ell ^2 (R/I,H_I)\).
Next we define \(S_a\) for \(a\in R^{\times }\). By Marcelo and Machiel [14, Lemma 1.11] there exists a multiplicative cross section of the quotient \(R^{\times }\rightarrow R^{\times }/R^*\) and thus we have a homomorphism \(a \mapsto \tilde{a}\) of \(R^{\times }\) into itself such that for each \(a \in R^{\times }\) there exists a unique \(g\in R^*\) with \(a = \tilde{a} g\).
For an ideal \(L\subset R\), we view \(\ell ^2(L/I,\;H_I)\) as the obvious subspace of \(\ell ^2(R/I, H_I)\) and we define \(E_L\) to be the orthogonal projection onto \(\bigoplus _{I\in \gamma ,I\subset L} \ell ^2(L/I,\;H_I)\).
It is easy to verify that \(S\) is a representation of the semigroup \(R^{\times }\) by isometries and that \(E\) is a family of projections representing the lattice of ideals of \(R\), such that \(U\), \(S\), and \(E\) satisfy the relations defining \(\mathfrak T [R]\). Hence there is a representation \(\mu _\tau \) of \(\mathfrak T [R]\) such that \(\mu _\tau (u^x) = U^x\), \(\mu _\tau (s_a) = S_a\) and \(\mu _\tau (e_I ) = E_I\).\(\square \)
Lemma 7.2
 (i)
if \(I \in \gamma \), then \(U^x\tilde{\delta }_IU^{x} = U^x E_I \mathcal E _I U^{x}\), the projection onto \(U^xH_I\);
 (ii)
if \(I \not \in \gamma \), then \(U^x\tilde{\delta }_IU^{x} = 0\); and
 (iii)the trace \(\tau \) is retrieved from \(\varphi _{\gamma , \tau }\) by conditioning to \(\tilde{\delta }_{J_\gamma }\):$$\begin{aligned} \tau (u^x s_g) =N(J_\gamma )^\beta \zeta _\gamma (\beta 1) \varphi _{\gamma ,\tau } (\tilde{\delta }_{J_\gamma }U^x S_g \tilde{\delta }_{J_\gamma }) \qquad x\in J_\gamma , \ \ g\in R^*. \end{aligned}$$
Proof
For part (i), notice that when \(I\in \gamma \), then \(\mathcal E _I = \tilde{\varepsilon }_I:= \lim _{n\rightarrow \infty } \mu _\tau (\varepsilon _{IP_1P_2\cdots P_n})\), then multiply by \(E_I\) and translate with \(x\in R/I\).
For part (ii), recall that if \(I\) and \(I^{\prime }\) are different ideals, then the projections \(\tilde{\delta }_I^x\) and \(\tilde{\delta }_{I^{\prime }}^{x^{\prime }}\) are mutually orthogonal. Since the Hilbert space \(H_\tau = \bigoplus _{I \in \gamma } \ell ^2 (R/I,H_I) \) is generated by the ranges of the projections \(\tilde{\delta }_I^x\) with \(I\in \gamma \) and \(x\in R/I\), it follows that \(\tilde{\delta }_{I^{\prime }}^{x^{\prime }} =0\) whenever \(I^{\prime } \not \in \gamma \).
Finally, notice that \(H_{J_\gamma }\) viewed as a subspace of \(H_\tau \) is invariant for the action of \(C^*(J_\gamma \rtimes R^*)\) and, by construction, the restriction of \(\mu _\tau \) to \(C^*(J_\gamma \rtimes R^*)\) and to this subspace is the GNS representation of \(\tau \), with cyclic vector \(\xi _{J_\gamma }\). Since \(\tilde{\delta }_{J_\gamma }=E_{J_\gamma } \mathcal E _{J_\gamma }\) is the projection onto \(H_{J_\gamma }\), the sum in Eq. (9) has only one term, giving the identity in part (iii).\(\square \)
It turns out that to parametrize the \(\beta \)KMS states in the region \(\beta >2\) all we need to do is combine states constructed from different ideal classes.
Theorem 7.3
Suppose \(\beta >2\) and choose a fixed reference ideal \(J_\gamma \in \gamma \) for each \(\gamma \) in the class group \(\Gamma \) of \(K\). For each tracial state \(\tau \) of \(\bigoplus _\gamma C^*(J_\gamma \rtimes R^*)\) write \(\tau = c_\gamma \tau _\gamma \) as a convex linear combination of traces on the components and define \(\varphi _\tau := \sum _\gamma c_\gamma \varphi _{\gamma ,\tau _\gamma }\) using Eq. (9). Then the map \(\tau \mapsto \varphi _\tau \) is a continuous affine isomorphism of the Choquet simplex of tracial states of \(\bigoplus _\gamma C^*(J_\gamma \rtimes R^*)\) onto the simplex of \(\beta \)KMS states for \(\mathfrak T [R]\).
Proof
Suppose \(\varphi \) is a \(\beta \)KMS state and let \(\mathfrak T [R]\) be represented on \(H_\varphi \) in the GNSconstruction for \(\varphi \). As usual, we denote by \(\tilde{\varphi }\) the vector state extension of \(\varphi \) to \(\mathcal L (H_\varphi )\), and we also write \(\pi _\varphi (u^x) = U^x\), \(\pi _\varphi (s_a) = S_a\) for simplicity of notation.
Let \(F\) denote the orthogonal complement of \(\bigoplus _{I} \tilde{\varepsilon }_I (H_\varphi )\) and \(\psi \) the restriction of \(\tilde{\varphi }\) to \(\pi _\varphi (\mathfrak T [R])_F\). Since \(\psi \) is again a \(\beta \)KMS functional (see [3], 5.3.4 and 5.3.29) and since the \(F \tilde{\varepsilon }_{J_\gamma } F=0\), the above identity applied to \(\psi \) shows that \(\psi (F)=0\) and thus \(F=0\), proving \(\bigoplus _{I} \tilde{\varepsilon }_I (H_\varphi )=H_\varphi \).
The map \(\varphi \mapsto \tau \) is clearly affine and continuous in the weak*topology, and since the spaces of traces and of \(\beta \)KMS states are compact Hausdorff, the map is a homeomorphism.\(\square \)
Remark 7.4
 (1)
Our parameter space of traces is obviously not canonical because it depends on the arbitrary choice of representative ideals \(J_\gamma \) in each class. However, the traces are determined up to canonical isomorphisms of the underlying C*algebras, as discussed at the beginning of the section.
 (2)The \(\beta \)KMS states can be evaluated explicitly on products of the form \(s_a^* e^z_J u^y s_b\); since these have dense linear span, this characterizes \(\varphi _\tau \). Assume first \(\tau \) is supported on a single ideal class \(\gamma \in \Gamma \). By (9) we may assume \(a^{1}b = g\in R^*\), for otherwise \(\varphi _\tau (s_a^* e^z_J u^y s_b) =0\). ThenThe nontrivial contributions come from terms with$$\begin{aligned} \varphi _{\gamma ,\tau } (s_a^* e^z_J u^y s_b)&= \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma }\sum _{x\in R/I} (S_a^* E^z_J U^y S_b \Delta ^\beta U^x\xi _I\,\,U^x\xi _I)\\&= \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma }\sum _{x\in R/I} (U^{x}S_a^* E^z_J U^y S_bU^x \Delta ^\beta \xi _I\,\,\xi _I)\\&= \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma }\sum _{x\in R/I} N(I)^{\beta }(U^{x}S_a^* E^z_J U^y S_bU^x \xi _I\,\,\xi _I)\\&= \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma }\sum _{x\in R/I} N(I)^{\beta }(S_a^* U^{ax}E^z_J U^{y + agx}S_g S_a\xi _I\,\,\xi _I)\\&= \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma }\sum _{x\in R/I} N(I)^{\beta }( E^{zax}_J U^{y+ agx ax}S_g \xi _{aI}\,\,\xi _{aI}). \end{aligned}$$Thus, recalling that \(\xi _{aI}\) is the cyclic vector for the GNS representation of \(\tau _I\) (the notation is from the construction leading up to Lemma 7.1), the sum reduces to

\(zax \in J\),

\(y+ a(g1)x\in {aI}\) and

\(aI \subset J\).
where \(P_I:= \{x\in R/I\, \big  \, axz\in J/I, \, y+ax(g1) \in I\}\). If we now start with a trace \(\tau = \sum _{\gamma \in \Gamma } c_\gamma \tau _\gamma \), then the values of the corresponding \(\beta \)KMS state are given by$$\begin{aligned} \varphi _{\gamma ,\tau } (s_a^* e^z_J u^y s_b) = \frac{1}{\zeta _\gamma (\beta 1)}\sum _{I\in \gamma ,\, aI \subset J}\sum _{x\in P_I} N(I)^{\beta }\tau _{aI}( u^{y+ ax(g1) }s_g) \end{aligned}$$$$\begin{aligned} \varphi (s_a^* e^z_J u^y s_b) = \sum _{\gamma \in \Gamma }\sum _{I\in \gamma ,\, aI \subset J}\sum _{x\in P_I} \frac{c_\gamma N(I)^{\beta }}{\zeta _\gamma (\beta 1)} \tau _{\gamma , \,aI}( u^{y+ a(g1)x }s_g). \end{aligned}$$ 
 (3)The \(\infty \)KMS states are, by definition, the weak* limits as \(\beta \rightarrow \infty \) of \(\beta \)KMS states, and they too can be computed explicitly, by taking limits in the above formula. Notice that \(\frac{N(I)^{\beta }}{\zeta _\gamma (\beta 1)} \rightarrow 0\) as \(\beta \rightarrow \infty \), except when \(I\) is normminimizing in its class, in which case the limit is \(k_\gamma ^{1}\) (with \(k_\gamma \) the number of normminimizing ideals in the class \(\gamma \)). Thus, \(\infty \)KMS states are still indexed by traces \(\tau =\sum _\gamma c_\gamma \tau _\gamma \) of \(\bigoplus _\gamma C^*(J_\gamma \rtimes R^*)\), and are given bywhere the sum is now over the subset \(\underline{\gamma } \) of normminimizing ideals in \(\gamma \).$$\begin{aligned} \varphi (s_a^* e^z_J u^y s_b) = \sum _{\gamma \in \Gamma }\sum _{I\in \underline{\gamma }, \, aI \subset J}\sum _{x\in P_I} {c_\gamma k_\gamma ^{1}} \tau _{\gamma , \,aI}( u^{y+ a(g1)x }s_g). \end{aligned}$$
Remark 7.5
As a much simpler “toy model” for the dynamical system \((\mathfrak T [R], (\sigma _t))\) we can also consider the Toeplitz algebra \(\mathfrak T [R^{\times }]\) associated with the multiplicative semigroup \(R^{\times }\) of \(R\), i.e. the C*algebra generated by the left regular representation of this semigroup. It is generated by isometries \(s_a\), \(a \in R^{\times }\) and carries an analogous oneparameter automorphism group \((\sigma ^\times _t)\) defined by \(\sigma ^\times _t(s_a) = N(a)^{it}s_a\). Since \(R^\times \) is a split extension of \(R^\times /R^*\) by \(R^*\), [14, Lemma 1.11], we see that \(\mathfrak T [R^\times ]\) is the tensor product of \(C^*(R^*)\) and the Toeplitz algebra \(\mathfrak T [R^\times /R^*]\) for the semigroup \(R^\times /R^*\) of principal integral ideals. In the case where \(R\) is a principal ideal domain, \(\mathfrak T [R^{\times }]\) is then simply an infinite tensor product of the ordinary Toeplitz algebras (i.e. universal C*algebras generated by a single isometry) generated by the isometries associated to the primes in \(R\), and of \(C^*(R^*)\). In this case the situation is nearly trivial. An easy exercise shows that the KMSstates for each \(\beta >0\) are labeled by the states of \(C^*(R^*)\).
However, in the case of a nontrivial class group, we obtain a nontrivial C*dynamical system, essentially, because there is an ‘interaction’ between the classes. The methods and results of the last two sections (including Theorem 5.6) immediately lead to a determination of its KMS structure. One finds that for \(\beta = 0\) there is a family of \(0\)KMS states (\(\sigma \)invariant traces) indexed by the \(\sigma \)invariant states on \(C^*(K^\times )\) (such a state has to factor through the quotient of \(\mathfrak T [R]\) where each of the generators \(s_a\) becomes unitary—this quotient is exactly \(C^*(K^\times )\)). For each \(\beta \) in the range \(0 < \beta \le 1\) the \(\beta \)KMS states correspond exactly to the states of \(C^*(R^*)\) [(there is a unique \(\beta \)KMS state on \(\mathfrak T [R^\times /R^*]\) which can be combined with an arbitrary state on the tensor factor \(C^*(R^*)\)]. For each \(\beta \) in the range \(1 < \beta < \infty \) the simplex of KMS states splits in addition over the class group \(\Gamma \). Thus the KMS states in that range are labeled by the states of \(C^*(R^*\times \hat{\Gamma })\).
We note that it is known that the class group \(\Gamma \) for \(K\) is determined already by the semigroup \(R^\times \). In fact \(\Gamma \) coincides with the semigroup class group defined by the ideals in this semigroup (i.e. the subsets invariant under multiplication by all elements), cf. [7, section 2.10].
8 Ground states
Proposition 8.1
 (1)
\(\varphi \) is a ground state;
 (2)
for all \(d\in \bar{\mathcal D }\), \(a,b\in R^\times \), \(x\in R\) and \(w\in \mathfrak T [R]\) we have \(\varphi ( w\, s_a^* d u^x s_b) = 0 \), whenever \(N(a)> N(b)\);
 (3)
for \(a,b\in R^\times \), \(x\in R\), we have \(\varphi (s_b^*u^xs_as_a^* u^{x}s_b)=0\), whenever \(N(a)> N(b)\)\((\)note that the expression under \(\varphi \) depends on \(x\) only via its image in \(R/aR)\);
Proof
We will see that the ground states on \(\mathfrak T [R]\) are supported on projections corresponding to what we call “normminimizing ideals”. We say that an ideal \(I\) in \(R\) is normminimizing if for any other ideal \(J\) in the same ideal class we have \(N(I)\le N(J)\). The use of normminimizing ideals was suggested by work in preparation by Laca–van Frankenhuijsen.
Recall from Lemma 4.15 (a) that every ideal \(I\) in \(R\) can be written in the form \(\frac{a}{b}R\cap R\) with \(a,b\in R^\times \).
Lemma 8.2
 (i)
If a product \(J = I L\) is normminimizing, then so are \(I\) and \(L\).
 (ii)
The prime ideals that are normminimizing generate the ideal class group.
 (iii)
If \(I = \frac{a}{b}R\cap R\) is normminimizing, then \(N(a)\le N(b)\).
Proof
The proof of part (i) is obvious and (ii) follows easily from (i). To prove (iii) observe that for each \(I=\frac{a}{b}R\cap R\), the integral ideal \(I^{\prime }=\frac{b}{a}I=R\cap \frac{b}{a}R\) is in the same class and \(N(I^{\prime })=N(b)N(a)^{1}N(I)\). Thus, if \(I\) is norm minimizing, necessarily \(N(b)N(a)^{1}\ge 1\).\(\square \)
Lemma 8.3
Let \(\varphi \) be a ground state of \(\mathfrak T [R]\). Then \(\varphi (e_I^x)=0\) for each ideal \(I\) in \(R\) which is not normminimizing and for each \(x\in R/I\).
Proof
If \(I\) and \(J\) are two ideals in the same ideal class, then there exist integers \(a\) and \(b\) in \(R^\times \) such that \(bI = aJ\), so \(e_I = s_b^* s_a e_J s_a^* s_b\). Assuming that \(J\) is normminimizing but \(I\) is not, then \(N(a)> N(b)\) by Lemma 8.2(iii), so we may use part (2) of Proposition 8.1 on the product \((u^x s_b^* s_a e_J) \, (s_a^* s_b u^{x})=e_I^x\) to finish the proof.
\(\square \)
In particular, the above proposition implies that \(\varphi (e_P^x)=0\) for each prime ideal \(P\) which is not normminimizing and for each \(x\in R/P\). Thus \(\varphi (\varepsilon _P)=\varphi (1 f_P)=1\) for such ideals. To take advantage of this feature, we will order the prime ideals in \(R\) in such a way that \(P_1,\ldots , P_k\) are normminimizing while all the other prime ideals \(P_{k+1},P_{k+2},\ldots \) are not. By Lemma 8.2(ii) the (finite) set \(\underline{\mathcal I _k}\) of normminimizing ideals in the semigroup \(\mathcal I _k\) generated by the \(P_1,\ldots , P_k\) is in fact the finite set of all normminimizing ideals of \(R\). The projection \(\varepsilon _{\underline{\mathcal I _k}}:=\sum _{I\in \underline{\mathcal I _k}}\varepsilon _{IP_1\cdots P_k}\) corresponding to the normminimizing ideals will be the key to our characterization of ground states.
Lemma 8.4
Let \(\varphi \) be a ground state of \(\mathfrak T [R]\) and assume \(n > k\) so that \(P_1,\ldots , P_k\) are normminimizing while \(P_{k+1},P_{k+2},\ldots , P_n\) are not. If \(\varepsilon _{\underline{\mathcal I _k}}:=\sum _{I\in \underline{\mathcal I _k}}\varepsilon _{IP_1\cdots P_k}\), then \(\varphi (\varepsilon _{\underline{\mathcal I _k}}\varepsilon _{P_{k+1}P_{k+2}\cdots P_n})=1\).
Proof
Recall the minimal projections \(\delta ^x_{I,n}\in \mathcal D _n\), for \(I\in \mathcal I _n,\,x\in R/I\), introduced in Sect. 4.
We will now consider \(\mathfrak T [R]\) in its universal representation. Thus let \(S\) be the state space of \(\mathfrak T [R]\) and let \(\pi _S=\bigoplus _{f\in S}\pi _f\) be its universal representation on the Hilbert space \(H_S=\bigoplus _{f\in S}H_f\). We will from now on assume that \(\mathfrak T [R]\) is represented via \(\pi _S\) and we will omit the \(\pi _S\) from our notation.
If \(\varphi \) is a state of \(\mathfrak T [R]\), we denote by \(\tilde{\varphi }\) its unique normal extension to the von Neumann algebra \(\mathfrak T [R]^{\prime \prime }\) generated by \(\mathfrak T [R]\).
We write \(\tilde{\delta }_I\), \(\tilde{\delta }_I^x\), \(\tilde{\varepsilon }_I\) for the strong limits, as \(n\rightarrow \infty \), of the monotonously decreasing sequences of projections \(\delta _{I,n}\), \(\delta _{I,n}^x\) and \(\varepsilon _{IP_1P_2\ldots P_n}\), respectively (recall that \(\delta _{I,n}:= \delta _{I,n}^0\)).
In the representation \(\mu \) used in Sect. 4, the projection \(\tilde{\delta }_I^x\) is represented by the projection onto the onedimensional subspace of \(\ell ^2(R/I)\) corresponding to \(x\in R/I\). It is therefore nonzero.
Proposition 8.5
A state \(\varphi \) of \(\mathfrak T [R]\) is a ground state if and only if \(\tilde{\varphi } (E)=1\).
Proof
If \(\varphi \) is a ground state, then \(\tilde{\varphi }(E) =1\) follows immediately from Lemma 8.4 because \(\tilde{\varphi }\) is normal.
If, conversely, \(\tilde{\varphi } (E)=1\), then \(\varphi (w)=\tilde{\varphi }(w)=\tilde{\varphi }(EwE)\) for each \(w\in \mathfrak T [R]\). In order to show that condition (3) in Proposition 8.1 is satisfied, i.e. that \(\tilde{\varphi }(Es_b^*u^xs_as_a^*u^{x}s_bE)=0\) whenever \(N(a)>N(b)\), it suffices to show that \(\delta _I^ys_b^*u^xs_as_a^*u^{x}s_b\delta _I^{y}=0\) for all \(I\in \underline{\mathcal I _k}\) and \(y\in R/I\), whenever \(N(a)>N(b)\). This amounts to showing that \(\delta _{bI}^ys_as_a^*\delta _{bI}^{y}=0\) whenever \(N(a)>N(b)\). However, by Eq. (10), this last expression can be nonzero only if \(bI\subset aR\). This inclusion implies that \(I\subset \frac{a}{b}R\cap R\), i.e. that the ideal \(\frac{a}{b}R\cap R\) divides \(I\). Since \(I\) is normminimizing, \(\frac{a}{b}R\cap R\) then has to be normminimizing, too, and \(N(a)\le N(b)\) by Lemma 8.2(iii).
\(\square \)
Lemma 8.6
Let \(I, J\in \mathcal I _n\) and let \(a,b,a^{\prime },b^{\prime }\in R\) such that \(aI=bJ\) and \(a^{\prime }I=b^{\prime }J\). Then there is \(g\in R^*\) such that \(s_{b^{\prime }}^*s_{a^{\prime }}=s_gs_b^*s_a\). The operators \(s_{a^{\prime }}^*s_{b^{\prime }}\tilde{\delta }_I\) and \(s_b^*s_a\tilde{\delta }_I\) are partial isometries with support \(\tilde{\delta }_I\) and range \(\tilde{\delta }_J\). If \(I,J,L\) are three ideals in \(\mathcal I _n\) and \(aI=bJ=cL\), then \(s_c^*s_b\tilde{\delta }_J s_b^*s_a\tilde{\delta }_I=s_c^*s_a\tilde{\delta }_I\)
Proof
We have \((a/b)I=J=(a^{\prime }/b^{\prime })I\) whence \(a^{\prime }/b^{\prime }=ga/b\) for some \(g\in R^*\). Thus \(gab^{\prime }=a^{\prime }b\) and \(s_gs_as_{b^{\prime }}=s_{a^{\prime }}s_b\). Multiplying this from the left by \(s_a^*s_{a^{\prime }}^*\) gives the first assertion (note that \(s_g\) and \(s_g^*\) commute with \(s_a,s_{a^{\prime }}\)). The second assertion then follows from Lemma 4.11. Finally, \(s_c^*s_bs_b^*s_a\tilde{\delta } _I=s_c^*s_a\tilde{\delta }_I\) from Eq. (10) and the fact that \(aI\subset bR\).\(\square \)
Proposition 8.7
Proof
We use the partition of \(E\) as a sum of the projections \(\tilde{\delta }_I^x\), \(I\in \underline{\mathcal I _k}\), \(x\in R/I\).
Theorem 8.8
The ground states of \(\mathfrak T [R]\) are exactly the states of the form \(\varphi (w)=\psi (EwE)\) where \(\psi \) is an arbitrary state of \(E\mathfrak T [R]E\cong \bigoplus _\gamma M_{k_\gamma N(J_\gamma )}(C^*(J_\gamma \rtimes R^*))\).
Proof
This is immediate from Propositions 8.5 and 8.7.\(\square \)